理論計算機科学における 数学の応用: 擬似ランダムネス

東北大学数学科で2024年5月27日に行った談話会での発表資料
keyword: 擬似ランダムネス, エクスパンダーグラフ

Transcript

1. ཧ࿦ܭࢉػՊֶʹ͓͚Δ ਺ֶͷԠ༻: ٖࣅϥϯμϜωε ਗ਼ਫ ৳ߴ (౦ژ޻ۀେֶ) ஊ࿩ձ (2024೥5݄27೔@౦๺େֶ)
2. •ཧ࿦ܭࢉػՊֶͰ͸(૝૾Ҏ্ʹ)਺ֶͷ֓೦͕෯޿͘ొ৔ •਺ֶͱཧ࿦ܭࢉػՊֶͷڞ௨෦෼ͷҰͭ: ٖࣅϥϯμϜੑ ൃදͷ֓ཁ 2
3. •ཧ࿦ܭࢉػՊֶͰ͸(૝૾Ҏ্ʹ)਺ֶͷ֓೦͕෯޿͘ొ৔ •਺ֶͱཧ࿦ܭࢉػՊֶͷڞ௨෦෼ͷҰͭ: ٖࣅϥϯμϜੑ •ʮ͜Ε͕͜Μͳͱ͜Ζʹڞ௨఺͕͋Δͷ͔ʂ(ڻ)ʯͱࢥͬͯ΄͍͠ ൃදͷ֓ཁ 3
4. •ཧ࿦ܭࢉػՊֶͰ͸(૝૾Ҏ্ʹ)਺ֶͷ֓೦͕෯޿͘ొ৔ •਺ֶͱཧ࿦ܭࢉػՊֶͷڞ௨෦෼ͷҰͭ: ٖࣅϥϯμϜੑ •ʮ͜Ε͕͜Μͳͱ͜Ζʹڞ௨఺͕͋Δͷ͔ʂ(ڻ)ʯͱࢥͬͯ΄͍͠ •ʮงғؾʯΛ঺հ ‣ ݫີͳఆٛ΍ূ໌͸ׂѪ ൃදͷ֓ཁ 4
5. ೔ৗੜ׆ʹ͓͚Δܭࢉػͷར༻ 5
6. •ܭࢉػͷཧ࿦తͳೳྗ΍ͦͷݶքΛ਺ֶΛ࢖ͬͯղ໌ (Ԡ༻਺ֶ) ‣ ࠷దԽΞϧΰϦζϜ ‣ ࠔ೉ੑ (ܭࢉྔԼք; ༧૝) ‣ άϥϑΞϧΰϦζϜ ‣ ҉߸, ֶशཧ࿦ ‣ Ϛϧίϑ࿈࠯ ‣ ਺஋ܭࢉ ‣ ྔࢠΞϧΰϦζϜ ‣ ෼ࢄΞϧΰϦζϜ ‣ σʔλߏ଄ ‣ etc 𝖯 ≠ 𝖭 𝖯 ཧ࿦ܭࢉػՊֶ (Theoretical Computer Science) 6
7. TCSͱ(७ਮ)਺ֶͷܨ͕Γ 7 ର਺ιϘϨϑෆ౳ࣜ ϥϯμϜ΢ΥʔΫͷղੳ Green—Taoͷఆཧ ऑֶशثͷϒʔεςΟϯά Kazhdanͷੑ࣭ (T) ୤ཚ୒Խ ޡΓగਖ਼ූ߸ Bogolyubov—Ruzsaͷิ୊ ࠷ѱ͔࣌Βฏۉ࣌΁ͷؼண ପԁۂઢ҉߸ ପԁۂઢ ଟ༷ମͷCheegerఆ਺ ނো଱ੑωοτϫʔΫ άϥεϚϯଟ༷ମ 2-to-2༧૝ Hilbert’s Nullstellensatz Combinatorial Nullstellensatz ΞΠσΞΛഈआ ൓ྫͷߏ੒ ৽ͨͳ໰୊ઃఆ Connes ͷຒΊࠐΈ༧૝ MIP*=REఆཧ Baum—Connes༧૝ ηΩϡϦςΟ ฏۉ࣌ܭࢉྔ ੑ࣭ݕࠪ
8. TCSͱ(७ਮ)਺ֶͷܨ͕Γ 8 ର਺ιϘϨϑෆ౳ࣜ ϥϯμϜ΢ΥʔΫͷղੳ Green—Taoͷఆཧ ऑֶशثͷϒʔεςΟϯά ୤ཚ୒Խ ޡΓగਖ਼ූ߸ Bogolyubov—Ruzsaͷิ୊ ࠷ѱ͔࣌Βฏۉ࣌΁ͷؼண ପԁۂઢ҉߸ ପԁۂઢ ଟ༷ମͷCheegerఆ਺ ނো଱ੑωοτϫʔΫ άϥεϚϯଟ༷ମ 2-to-2༧૝ Hilbert’s Nullstellensatz Combinatorial Nullstellensatz ΞΠσΞΛഈआ ൓ྫͷߏ੒ ৽ͨͳ໰୊ઃఆ Connes ͷຒΊࠐΈ༧૝ MIP*=REఆཧ Baum—Connes༧૝ ηΩϡϦςΟ ฏۉ࣌ܭࢉྔ Kazhdanͷੑ࣭ (T) ٖࣅϥϯμϜੑ (pseudorandomness)
9. •ٖࣅཚ਺ͷੜ੒ ‣ ཚ਺Λ࢖͏৔໘ : ϥϯμϜ΢ΥʔΫ, MCMC, ֬཰తޯ഑๏, etc. ‣ ࣮ࡍͷܭࢉػͰ͸ٖࣅཚ਺Λ࢖ͬͯΔ - : ԿΒ͔ͷؔ਺ - Ͱٖࣅཚ਺Λͨ͘͞Μੜ੒ ( Λγʔυ஋ͱݺͿ) ‣ ༗໊ͳؔ਺ : ϝϧηϯψπΠελ, ઢܗ߹ಉ๏, etc ‣ ྑ࣭ͳ(=࣍ͷग़໨͕༧ଌͰ͖ͳ͍)ٖࣅཚ਺͕ཉ͍͠ f: {0,1}32 → {0,1}32 s → f(s) → f(f(s)) → ⋯ s ٖࣅཚ਺ 9
10. ٖࣅཚ਺ 10 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=503M62000000001) Χ ジ ϊ؅ཧҕһձؔ܎ಛఆෳ߹؍ޫࢪઃ۠Ҭ੔උ๏ࢪߦنଇ ୈ176৚ผද (H30੍ఆ)
11. ٖࣅཚ਺ 11 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=503M62000000001) Χ ジ ϊ؅ཧҕһձؔ܎ಛఆෳ߹؍ޫࢪઃ۠Ҭ੔උ๏ࢪߦنଇ ୈ176৚ผද (H30੍ఆ) •࣭͕ѱ͍ٖࣅཚ਺ͷࣄྫ (࣮࿩) ‣ 2006೥ʹൃച͞Εͨ๭ήʔϜιϑτʹͯʮμΠεͷ࣍ͷग़໨ͷۮح͕ਪଌͰ͖Δʯ ͱ͍͏க໋తͳόά͕ݟ͔ͭΓɺ঎඼ճऩʹࢸͬͨ.
12. •ٖࣅཚ਺ʹཉ͍͠ੑ࣭ ‣ Ұ༷ϥϯμϜͳ ʹର͠, ͕Ұ༷ϥϯμϜ - ݪཧతʹ͸ෆՄೳ ( ͕ܾ·Δͱ ΋ܾ·Δ͔Β) s (s, f(s)) s f(s) ٖࣅϥϯμϜੑ 12
13. •ٖࣅཚ਺ʹཉ͍͠ੑ࣭ ‣ Ұ༷ϥϯμϜͳ ʹର͠, ͕Ұ༷ϥϯμϜ - ݪཧతʹ͸ෆՄೳ ( ͕ܾ·Δͱ ΋ܾ·Δ͔Β) •ٖࣅϥϯμϜωε ‣ Ұ༷ϥϯμϜͳ ʹର͠, ͕Ұ༷ϥϯμϜͬΆ͘ݟ͑Δ ‣ Ұ༷෼෍ͱࣝผͰ͖ͳ͍Α͏ͳ෼෍ s (s, f(s)) s f(s) s (s, f(s)) ٖࣅϥϯμϜੑ 13 ৚݅Λ؇࿨
14. ෼෍ͷࣝผ 14 01010101010101010101 01000111001111001111 ͋Δ෼෍ ͔Βੜ੒͞Εͨ20จࣈ 𝒟 ࣝผऀ A ͬͪ͜͸Ұ༷ϥϯμϜ͡Όͳ͍ ͬͪ͜͸Ұ༷ϥϯμϜͰ͋Ζ͏ Ұ༷෼෍ ͔Βੜ੒͞Εͨ20จࣈ 𝒰
15. ෼෍ͷࣝผ 15 01010101010101010101 01000111001111001111 ͋Δ෼෍ ͔Βੜ੒͞Εͨ20จࣈ 𝒟 Ұ༷෼෍ ͔Βੜ੒͞Εͨ20จࣈ 𝒰 ࣝผऀ A ؔ਺ : 20จࣈ 0 or 1 A ↦ ࣝผऀ ͕ ͱ Λ -ࣝผ ͢Δ A 𝒟 𝒰 ε def ⟺ Pr[A( 𝒟 ) = 1] − Pr[A( 𝒰 ) = 1] > ε
16. •෼෍ Λ ʮ01010101010ʯʮ10101010101ʯͷͲͪΒ͔͕֬཰ Ͱग़ݱ •ࣝผऀ : ‣ 0ͱ1͕ަޓͳΒ͹1, ͦ͏Ͱͳ͍ͳΒ0Λग़ྗ • ͸0.999-ࣝผ ‣ ‣ 𝒟 1/2 A(s) A Pr[A( 𝒟 ) = 1] = 1 Pr[A( 𝒰 ) = 1] = 2/211 ≈ 0.001 ෼෍ͷࣝผ (ྫ) 16
17. •ٖࣅϥϯμϜωε ‣ ੍ݶ͞ΕͨࣝผऀͷΫϥε Λߟ͑Δ (ଟ߲ࣜ࣌ؒΞϧΰϦζϜͳͲ) 𝒜 ٖࣅϥϯμϜੑ 17 ෼෍ ͸ ʹରͯ͠ -ٖࣅϥϯμϜ Ͱ͋Δ ೚ҙͷ ͕ ͱ Λ -ࣝผ͠ͳ͍ 𝒟 𝒜 ε def ⟺ A ∈ 𝒜 𝒟 𝒰 ε શ஌શೳͷࣝผऀ ੍ݶ͞Εͨࣝผऀ 011010100 ૉ਺൪໨ͷจࣈ͕1ͩʂ 0ͱ1͕ަޓ͡Όͳ͍͔Β Ұ༷ϥϯμϜ͔ͳ͊
18. •ٖࣅϥϯμϜωε ‣ ੍ݶ͞ΕͨࣝผऀͷΫϥε Λߟ͑Δ (ଟ߲ࣜ࣌ؒΞϧΰϦζϜͳͲ) 𝒜 ٖࣅϥϯμϜੑ 18 ෼෍ ͸ ʹରͯ͠ -ٖࣅϥϯμϜ Ͱ͋Δ ೚ҙͷ ͕ ͱ Λ -ࣝผ͠ͳ͍ 𝒟 𝒜 ε def ⟺ A ∈ 𝒜 𝒟 𝒰 ε ✓ ܭࢉྔతٖࣅϥϯμϜੑ = ͕ޮ཰తͳΞϧΰϦζϜͷ଒ (ྫ: ଟ߲ࣜ࣌ؒΞϧΰϦζϜ) ✓ ૊߹ͤ࿦తٖࣅϥϯμϜੑ = ͕૊߹ͤతʹఆ·Δؔ਺ͷ଒ (ྫ: ࣍ଟ߲ࣜ) 𝒜 𝒜 d
19. ܭࢉྔతٖࣅϥϯμϜੑ ฏۉ࣌ܭࢉྔ
20. •ܭࢉྔ: ܭࢉͷෳࡶੑ (࣌ؒ, ۭؒetc) ΛਤΔई౓ ‣ ͕༩͑ΒΕͨͱ͖, Λܭࢉ͢Δखؒ͸ͲΕ͘Β͍͔? x f(x) ฏۉ࣌ࠔ೉ੑ 20
21. •ܭࢉྔ: ܭࢉͷෳࡶੑ (࣌ؒ, ۭؒetc) ΛਤΔई౓ ‣ ͕༩͑ΒΕͨͱ͖, Λܭࢉ͢Δखؒ͸ͲΕ͘Β͍͔? •࠷ѱ࣌ܭࢉྔ: ࠷ѱͳೖྗʹର͢ΔΞϧΰϦζϜͷڍಈ ‣ ͘͝গ਺ͷίʔφʔέʔεʹӨڹ͞Ε͏Δ ‣ “pessimism of worst-case analysis” [Frieze, McDiarmid, 1986] ‣ ༧૝ x f(x) 𝖯 ≠ 𝖭 𝖯 ฏۉ࣌ࠔ೉ੑ 21
22. •ܭࢉྔ: ܭࢉͷෳࡶੑ (࣌ؒ, ۭؒetc) ΛਤΔई౓ ‣ ͕༩͑ΒΕͨͱ͖, Λܭࢉ͢Δखؒ͸ͲΕ͘Β͍͔? •࠷ѱ࣌ܭࢉྔ: ࠷ѱͳೖྗʹର͢ΔΞϧΰϦζϜͷڍಈ ‣ ͘͝গ਺ͷίʔφʔέʔεʹӨڹ͞Ε͏Δ ‣ “pessimism of worst-case analysis” [Frieze, McDiarmid, 1986] ‣ ༧૝ •ฏۉ࣌ܭࢉྔ: ฏۉతͳೖྗʹର͢ΔΞϧΰϦζϜͷڍಈ ‣ গͳ͍ίʔφʔέʔεʹӨڹ͞Εʹ͍͘ x f(x) 𝖯 ≠ 𝖭 𝖯 ฏۉ࣌ࠔ೉ੑ 22
23. •ؔ਺ ͷܭࢉ͕ฏۉ࣌ࠔ೉ ‣ Ұ༷ϥϯμϜͳ ʹରͯ͠, ͷܭࢉ͕೉͍͠ ‣ ྫ: 10ܻͷϥϯμϜͳೋͭͷૉ਺ͷੵͷૉҼ਺෼ղ͸೉͍͠ (RSA҉߸) f x f(x) ฏۉ࣌ࠔ೉ੑ 23
24. •ؔ਺ ͷܭࢉ͕ฏۉ࣌ࠔ೉ ‣ Ұ༷ϥϯμϜͳ ʹରͯ͠, ͷܭࢉ͕೉͍͠ ‣ ྫ: 10ܻͷϥϯμϜͳೋͭͷૉ਺ͷੵͷૉҼ਺෼ղ͸೉͍͠ (RSA҉߸) •ฏۉ࣌ࠔ೉ͳؔ਺ ٖࣅཚ਺ੜ੒ث ‣ ʮ೉͍͠ʯͱ͍͏ωΨςΟϒͳੑ࣭ΛϙδςΟϒͳ݁ՌʹԠ༻ ‣ ͕ฏۉ࣌ࠔ೉ ͸೚ҙͷଟ߲ࣜ࣌ؒΞϧΰϦζϜʹͱٖͬͯࣅϥϯμϜ - Ұ༷ϥϯμϜͳ ʹରͯ͠ ͷܭࢉ͸೉͍͔͠Β f x f(x) ⇒ f ⟺ (s, f(s)) s f(s) ฏۉ࣌ࠔ೉ੑ 24 [Nisan, Wigderson, 1994]
25. ҉߸ 25 •҉߸ •๣डͨ͠ୈࡾऀʹ͍͔ͳΔ৘ใ΋࿙Εͯ͸͍͚ͳ͍ ‣ ҉߸จʹԿΒ͔ͷ౷ܭతಛ௃͕͋ͬͨΒඇࣗ໌ͳ৘ใ͕࿙ΕΔ ‣ ୈࡾऀʹͱͬͯϥϯμϜͳจࣈྻʹݟ͑Δ΂͖ Apple 0011101000100
26. ҉߸ 26 •҉߸ •๣डͨ͠ୈࡾऀʹ͍͔ͳΔ৘ใ΋࿙Εͯ͸͍͚ͳ͍ ‣ ҉߸จʹԿΒ͔ͷ౷ܭతಛ௃͕͋ͬͨΒඇࣗ໌ͳ৘ใ͕࿙ΕΔ ‣ ୈࡾऀʹͱͬͯϥϯμϜͳจࣈྻʹݟ͑Δ΂͖ - ୈࡾऀ͸੍ݶ͞ΕͨܭࢉೳྗΛ༗͢ΔͱԾఆ Apple 0011101000100
27. ҉߸ 27 •҉߸ •༗໊ͳ҉߸ํࣜ ‣ RSA҉߸ (ϥϯμϜͳڊେͳೋͭͷૉ਺ͷੵͷૉҼ਺෼ղͷࠔ೉ੑΛԾఆ) ‣ ֨ࢠ҉߸ (ϥϯμϜͳ֨ࢠ্Ͱͷ࠷୹֨ࢠ໰୊ͷࠔ೉ੑΛԾఆ) ‣ ڀۃతͳ໨ඪ: ͷԾఆͷԼͰ҆શͳ҉߸Λ࡞Δ •ฏۉ࣌ࠔ೉ͳ໰୊ 㱺 ΄ͱΜͲͷೖྗͰਖ਼͘͠ղ͚ͳ͍ 𝖯 ≠ 𝖭 𝖯 Apple 0011101000100
28. ҉߸ 28 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=413M60000418002)
29. ҉߸ 29 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=413M60000418002) •๏ྩͷจݴʹʮૉҼ਺෼ղʯʮ༗ݶମʯʮପԁۂઢʯ •େਉ͕ೝΊΕ͹OKΒ͍͠
30. ૊߹ͤ࿦తٖࣅϥϯμϜੑ ΤΫεύϯμʔάϥϑ
31. • : -ਖ਼ଇάϥϑ ‣ શ௖఺ʹ઀ଓ͍ͯ͠Δล͕ ຊ • : ୯७ϥϯμϜ΢ΥʔΫͷભҠ֬཰ߦྻ ‣ ୯७ϥϯμϜ΢ΥʔΫ : Ұ༷ϥϯμϜͳྡ઀఺ʹભҠ ‣ G d d P P(u, v) = { 1 d 0 ΤΫεύϯμʔάϥϑ 31 3-ਖ਼ଇάϥϑ ͕ลΛͳ͢ {u, v} ͦΕҎ֎
32. • : -ਖ਼ଇάϥϑ ‣ શ௖఺ʹ઀ଓ͍ͯ͠Δล͕ ຊ • : ୯७ϥϯμϜ΢ΥʔΫͷભҠ֬཰ߦྻ ‣ ୯७ϥϯμϜ΢ΥʔΫ : Ұ༷ϥϯμϜͳྡ઀఺ʹભҠ ‣ G d d P P(u, v) = { 1 d 0 ΤΫεύϯμʔάϥϑ 32 3-ਖ਼ଇάϥϑ ͕ลΛͳ͢ {u, v} ͦΕҎ֎ ఆٛ (ΤΫεύϯμʔάϥϑ) ભҠ֬཰ߦྻ ͷݻ༗஋ ͕ Λຬͨ͢ͱ͖ -ΤΫεύϯμʔͱ͍͏. P 1 = λ1 ≥ … ≥ λn ≥ − 1 max{|λ2 |, |λn |} ≤ λ λ
33. • : -ਖ਼ଇάϥϑ ‣ શ௖఺ʹ઀ଓ͍ͯ͠Δล͕ ຊ • : ୯७ϥϯμϜ΢ΥʔΫͷભҠ֬཰ߦྻ ‣ ୯७ϥϯμϜ΢ΥʔΫ : Ұ༷ϥϯμϜͳྡ઀఺ʹભҠ ‣ G d d P P(u, v) = { 1 d 0 ΤΫεύϯμʔάϥϑ 33 3-ਖ਼ଇάϥϑ ͕ลΛͳ͢ {u, v} ͦΕҎ֎ ఆٛ (ΤΫεύϯμʔάϥϑ) ભҠ֬཰ߦྻ ͷݻ༗஋ ͕ Λຬͨ͢ͱ͖ -ΤΫεύϯμʔͱ͍͏. P 1 = λ1 ≥ … ≥ λn ≥ − 1 max{|λ2 |, |λn |} ≤ λ λ ؆୯ͷͨΊৗʹਖ਼ଇੑΛԾఆ ( ͸ରশͳͷͰ࣮ݻ༗஋Λ΋ͭ) P
34. •ϥϯμϜ΢ΥʔΫͷऩଋੑ ‣ άϥϑ͕͋Δ৚݅Λຬͨ͢ͱ ͷ෼෍͸ ্ͷҰ༷෼෍ʹҰҙʹऩଋ ‣ ऩଋͷ଎͞͸ͲΕ͘Β͍͔? Xt V ϥϯμϜ΢ΥʔΫͱΤΫεύϯμʔ 34 ೋ෦άϥϑ্Ͱ͸ऩଋ͠ͳ͍ ඇ࿈݁ͩͱऩଋઌ͕ҰҙͰ͸ͳ͍
35. •ϥϯμϜ΢ΥʔΫͷऩଋੑ ‣ άϥϑ͕͋Δ৚݅Λຬͨ͢ͱ ͷ෼෍͸ ্ͷҰ༷෼෍ʹҰҙʹऩଋ ‣ ऩଋͷ଎͞͸ͲΕ͘Β͍͔? •ΤΫεύϯμʔάϥϑ ‣ ϥϯμϜ΢ΥʔΫͷऩଋ͕଎͍άϥϑ Xt V ϥϯμϜ΢ΥʔΫͱΤΫεύϯμʔ 35 ೋ෦άϥϑ্Ͱ͸ऩଋ͠ͳ͍ ඇ࿈݁ͩͱऩଋઌ͕ҰҙͰ͸ͳ͍
36. •ૄͳΤΫεύϯμʔάϥϑ : ૄͳͷʹ࿈݁ੑ͕ڧ͍ ΤΫεύϯμʔͷݟͨ໨ 36 ؆୯ʹ෼அͰ͖ͦ͏ ෼அ͠ʹ͍͘
37. ‣ ௖఺਺ ( ) ‣ શͯͷ ͸ -ΤΫεύϯμʔ •ఆ਺ ʹରͯ͠ -ΤΫεύϯμʔ଒͸ଘࡏ͢Δ͔? ‣ ϥϯμϜʹ࡞Δͱਖ਼ͷ֬཰Ͱ (֬཰࿦తख๏) ‣ ϥϯμϜਖ਼ଇάϥϑ •ϥϯμϜωεΛ࢖Θͣʹߏ੒Ͱ͖Δ͔? (୤ཚ୒) → ∞ i → ∞ Gi λ λ < 1 λ λ = 2 d − 1 d + 0.01 ΤΫεύϯμʔάϥϑ 37 -ਖ਼ଇάϥϑͷ଒ ͸ -ΤΫεύϯμʔ଒Ͱ͋Δ d (Gi )i∈ℕ λ def ⟺ [Friedman, 2008]
38. •୅਺తͳߏ੒ ‣ έΠϦʔάϥϑ (܈ͷ࡞༻Λௐ΂Δॏཁͳಓ۩) ‣ Margulisͷߏ੒ (1973) … ‣ Lubotzky, Phillips, and Sarnak (1988) ‣ Margulis (1988) ‣ Morgenstern (1994) ‣ ࣍਺ ͕ಛผͳ৔߹ͷߏ੒ λ = 5 2 8 < 0.9 d ΤΫεύϯμʔ଒ͷߏ੒ 38
39. •୅਺తͳߏ੒ ‣ έΠϦʔάϥϑ (܈ͷ࡞༻Λௐ΂Δॏཁͳಓ۩) ‣ Margulisͷߏ੒ (1973) … ‣ Lubotzky, Phillips, and Sarnak (1988) ‣ Margulis (1988) ‣ Morgenstern (1994) ‣ ࣍਺ ͕ಛผͳ৔߹ͷߏ੒ λ = 5 2 8 < 0.9 d ΤΫεύϯμʔ଒ͷߏ੒ 39 ϥϚψδϟϯάϥϑ (“࠷దͳ”ΤΫεύϯμʔੑΛ΋ͭ) λ ≈ 2 d − 1 d
40. •૊߹ͤతͳߏ੒ ‣ ୅਺తͳߏ੒ͩͱ௚ײ (ͳͥΤΫεύϯμʔੑ͕੒Γཱͭͷ͔? )͕೉͍͠ ‣ Reingold, Vadhan, Wigderson (2002) - δάβάੵ ‣ Marcus, Spielman, Srinivasta (2015) - શͯͷೋ෦ϥϚψδϟϯάϥϑͷߏ੒ - ৫Γࠞͥଟ߲ࣜ (interlacing polynomial) •ະղܾ: ࣍਺7ͷϥϚψδϟϯάϥϑ଒ͷߏ੒ ΤΫεύϯμʔ଒ͷߏ੒ (ଓ) 40
41. •ཁૉ਺ ͷू߹ ʹର͠, ࣍ͷ ্ͷ෼෍ Λߟ͑Δ ‣ ্ͷ -ਖ਼ଇΤΫεύϯμʔάϥϑ Λߟ͑Δ ‣ ௖఺ ΛҰ༷ϥϯμϜʹબͿ ‣ Λ࢝఺ͱ͢Δ௕͞ ͷϥϯμϜ΢ΥʔΫͷܦ༝௖఺ Λग़ྗ n V Vℓ 𝒟 V d G = (V, E) u1 u1 ℓ − 1 (u1 , …, uℓ ) ٖࣅϥϯμϜੑ 41 u1 u2 uℓ
42. ٖࣅϥϯμϜੑ 42 άϥϑ ͕ -ΤΫεύϯμʔͳΒ͹, ͸ ʹର͠ -ٖࣅϥϯμϜ G λ 𝒟 𝒜 = {AS : S ⊆ V} (λ/4) ෦෼ू߹ ʹର͠, Λ S ⊆ V AS : Vℓ → {0,1} AS (u1 , …, uℓ ) = { 1 0 {u1 , …, uℓ } ∩ S ≠ ∅ ͦΕҎ֎ ఆཧ u1 u2 uℓ S = ్தͰ ͷ௖఺Λ௨ա͔ͨ͠Ͳ͏͔ AS (u1 , …, uℓ ) S
43. • ͷݩʹ͸ʮ͋ͨΓʯorʮ͸ͣΕʯ͕͋Δ ‣ গͳ͘ͱ΋ ݸͷʮ͋ͨΓʯ͕͋Δ - Ұ༷ϥϯμϜʹҾ͍ͨ௖఺͕͋ͨΔ֬཰ = •ಠཱҰ༷ϥϯμϜʹ ճ͘͡ΛҾ͘ ‣ ʮ͋ͨΓʯ͕ҰճҎ্ग़Δ֬཰ = ‣ ճҾ͚͹, 99%ͷ֬཰Ͱ͋ͨΓΛҾ͚Δ ‣ ͜ͷͱ͖, ϏοτͷϥϯμϜωε͕ඞཁ V δn δ ℓ 1 − (1 − δ)ℓ ℓ = 10/δ 10 log2 n δ Ԡ༻ 43
44. • ͷݩʹ͸ʮ͋ͨΓʯorʮ͸ͣΕʯ͕͋Δ ‣ গͳ͘ͱ΋ ݸͷʮ͋ͨΓʯ͕͋Δ - Ұ༷ϥϯμϜʹҾ͍ͨ௖఺͕͋ͨΔ֬཰ = • -ΤΫεύϯμʔάϥϑ্Ͱ ʹैͬͯ ճ͘͡ΛҾ͘ ‣ άϥϑͷ࣍਺ ͸ ʹґଘ͠ͳ͍ఆ਺ʹͰ͖Δ ‣ ͸ -ٖࣅϥϯμϜͳͷͰ98%ͷ֬཰Ͱʮ͋ͨΓʯΛҾ͘ ‣ ༻͍ͨϥϯμϜωε͸ V δn δ 0.04 𝒟 ℓ d n 𝒟 0.01 log2 n + 10 log2 d δ Ԡ༻ 44 u1 u2 uℓ ʮ͋ͨΓʯ ಠཱαϯϓϦϯάΑΓ΋গͳ͍ʂ
45. •PCPఆཧ •ޡΓగਖ਼ූ߸ •ٖࣅཚ਺ੜ੒ث •ϋογϡؔ਺ •ฏۉ࣌ܭࢉྔ ΤΫεύϯμʔάϥϑͷԠ༻ 45 [Charles, ’09] [Guruswami, Kabanets, ’08], [Goldreich, Impagliazzo, Levin, Venkatesan, Zuckerman, 90] [Goldreich, 00] [Sipser, Spielman, 96] [Dinur, 07]
46. •୯ମෳମͷΤΫεύϯμʔੑ ‣ ߴ࣍ݩΤΫεύϯμʔ •௕೥ͷະղܾ໰୊ͷղܾͷཱ໾ऀ ‣ ϚτϩΠυʹؔ͢ΔMihail—Vazirani༧૝ ‣ ہॴݕࠪՄೳޡΓగਖ਼ූ߸ͷߏ੒ •ຊߨٛͰ঺հ ۙ೥ͷಈ޲: ߴ࣍ݩΤΫεύϯμʔ 46
47. •ٖࣅϥϯμϜωε ‣ ७ਮ਺ֶͱTCSͷ྆ํʹݱΕΔ֓೦ ‣ ੔਺࿦, ܈࿦, زԿֶʹ΋Ԡ༻͞Ε͍ͯΔ(Β͍͠) •ΤΫεύϯμʔάϥϑ ‣ έΠϦʔάϥϑΛ࢖ͬͯߏ੒ ‣ ϥϯμϜωεΛʮઅ໿ʯͯ͘͠͡ΛҾ͘ํ๏ •ߴ࣍ݩΤΫεύϯμʔ ‣ ۙ೥ͷTCSͰϗοτͳ࿩୊ ‣ ߨٛͰ΍Γ·͢ʂ ·ͱΊ 47 [Lubotzky, ’12]